Unit 3: Lesson 3Linear Equations in Two Variables

Learning target:

Students use a table to find solutions to a given linear equation and plot the solutions on a coordinate plane.

Housekeeping• We discussed U3L2: Constant Rate yesterday• Did you complete the practice problems and submit your score

to the dropbox?

In this lesson, you will find solutions to a linear equation in two variables using a table, then plot the solutions as points on the coordinate plane. You will need graph paper in order to complete the Exercises and the Problem Set.

Emily tells you that she scored 32 points in a basketball game with only two- and three-point baskets (no free throws). How many of each type of basket did she score? Use the table below to organize your work.

Number of two-pointers Number of three-pointers

Let x be the number of two-pointers and y be the number of three-pointers that Emily scored. Write an equation to represent the situation.

An equation in the form of ax + by = c is called a linear equation in two variables, where a, b, and c are constants, and at least one of a and b are not zero. In this lesson, neither a nor b will be equal to zero.

In the Opening Exercise, what equation did you write to represent Emily’s score at the basketball game?

An equation of this form, ax + by = c, is also referred to as an equation in standard form. Is the equation you wrote for the example with Emily an equation in standard form?

In the equation ax + by = c, the symbols a, b, and c are constants. What, then, are x and y?

For example, −50x + y = 15 is a linear equation in x and y. As you can easily see, not just any pair of numbers x and y will make the equation true. Consider x = 1 and y = 2. Does it make the equation true?

What pairs of numbers did you find that worked for Emily’s basketball score? Did just any pair of numbers work? Explain.

A solution to the linear equation in two variables is an ordered pair of numbers (x, y) so that x and y makes the equation a true statement. The pairs of numbers that you wrote in the table for Emily are solutions to the equation 2x + 3y = 32 because they are pairs of numbers that make the equation true. The question becomes, how do we find an unlimited number of solutions to a given linear equation?

A strategy that will help us find solutions to a linear equation in two variables is as follows: We fix a number for x. That means we pick any number we want and call it x. Since we know how to solve a linear equation in one variable, then we solve for y. The number we picked for x and the number we get when we solve for y is the ordered pair (x, y), which is a solution to the two variable linear equation.

For example, let x = 5. Then, in the equation −50x + y = 15

Similarly, we can fix a number for y and solve for x. Let y = 10, then

-50x + y = 15

Now you pick a number for x or y and find the solution to the equation

-50x + y = 15

Find five solutions for the linear equation x + y = 3, and plot the solutions as points on a coordinate plane.

x Linear equation:x + y = 3

y

Find five solutions for the linear equation 2x − y = 10, and plot the solutions as points on a coordinate plane.

x Linear equation:2x – y = 10

y

Find five solutions for the linear equation x + 5y = 21, and plot the solutions as points on a coordinate plane.

x Linear equation:x + 5y = 21

y

Consider the linear equation x + y = 11.

Will you choose to fix values for x or y? Explain

Are there specific numbers that would make your computational work easier? Explain

Find five solutions to the linear equation x + y = 11, and plot the solutions as points on the coordinate plane.

x Linear equation:x + y = 11

y

At the store, you see that you can buy a bag of candy for $2 and a drink for $1. Assume you have a total of $35 to spend. You are feeling generous and want to buy some snacks for you and your friends. a. Write an equation in standard form to represent the number of bags of candy, x, and the number of drinks, y, that you can buy with $35.

Find five solutions to the linear equation from the first part, and plot the solutions as points on a coordinate plane.

x Linear equation: y

Wrap Up• A two-variable equation in the form of ax + by = c is known as a linear

equation in standard form. • A solution to a linear equation in two variables is an ordered pair (x, y) that

makes the given equation true. • We can find solutions by fixing a number for x or y, then solving for the

other variable. Our work can be made easier by thinking about the computations we will need to make before fixing a number for x or y. For example, if x has a coefficient of , we should select values for x that are multiples of 3.

What now???• Complete the U3L3 practice problems, check your answers by watching

the recording, and then submit your score to the dropbox.